Solution (source code)

= Solution

<Relativity of simultaneity> concerns separated events. Under Einstein-synchronized inertial coordinates, $\Delta t'=\gamma(\Delta t-v\Delta x/c^2)$. Two events with $\Delta t=0$ and $\Delta x\ne0$ are therefore generally nonsimultaneous in a relatively moving frame. The distinction is compatible with invariant coincidence at one event and with invariant temporal order for timelike-separated events. It is <spacelike separation> that permits differing coordinate order.

Conventionality enters when defining distant <clock> <Einstein synchronisation>[synchronization]. A send-and-return signal experiment measures a round-trip duration, but assigning a time to reflection at the distant <clock> requires a rule. Einstein's convention assigns equal outward and return times. Alternative admissible <Einstein synchronisation>[synchronizations] can reassign the one-way coordinate <Speed>[speeds] while preserving the observed two-way timings and local <clock> readings. These alternatives do not permit changing measured <proper time> at will or making any two causally ordered events operationally interchangeable.

The physical conclusion and convention claim should therefore be separated. No preferred inertial simultaneity relation is selected by the ordinary <Lorentz invariance>[Lorentz-invariant] structure. Once the standard <Einstein synchronisation>[synchronization] is fixed, relativity between <Inertial frame>[inertial frames] follows quantitatively. Reparameterizing distant times shows a conventional element in their coordinate representation; it does not eliminate every empirical constraint from relativity. Matter dynamics, causal cones and <clock> transport still restrict what counts as an equivalent description.

A claim that simultaneity is simply unreal overlooks its well-defined local and frame-relative uses. A claim that it is entirely conventional overlooks the invariant difference between causal and <spacelike separation>. The defensible account distinguishes those uses rather than expecting one slogan to settle them all.<On the Electrodynamics of Moving Bodies by Albert Einstein>[1905, sections 1–3, clock synchronization and coordinate transformation]{ref}